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Basically, its a 3rd party service specifically designed to be linked to. If you wanted more assurance, you could download and host it yourself from the official jQuery website: http://jquery.com/download/ - I'm not sure why it wouldn't be working for you on your own server though so maybe you should provide a link to you page for us to see.

C: see the link in Beverley's post. There is no guarantee, but the intention is for it to be available-- many more web designers than just you will have major issues if it is no longer available. (Every 5 years or so you might want to look into updating the websites to work smoothly with modern browsers, but that's a somewhat separate issue.)

A: Your site won't work. But Google has very little downtime, and they will make every effort to keep high demand files like that available.

In short: while it isn't guaranteed to always/forever be available, it's generally reliable. We use it; you can too.

B: It will work. There's nothing wrong with that, except for potential updates (which aren't strictly necessary) and freeing up space/bandwidth on your host.
Are you asking "why does it say/suggest it won't work?" (it will; it's just standard to use the hosted version), or "why isn't it working right now when I try it?"? If you have a problem with your page, please provide a link to it. My guess is that one of the file paths is incorrect so that it's looking for the script in the wrong place.

is the version you want. Notice that I have only 1.8 there. That will get you the lateset in the 1.8 series (1.8.2). Similarly you can use 1.6 for the latest in the 1.6 series (1.6.4), or just 1 for the latest in the 1 series (currently 1.9.1 - that may increase, the 1.9 series isn't necessarily finished yet, and 1.10 and 1.11 and 1.12, etc., even 1.9999 series are possibilities).

Or if you want a specific version, use it's full version, like 1.6.2 for example.

If you have some pages hosted on either an SSL (https:) or on a regular layer (http:), or both, the user's browser will fill in the correct protocol. And since Google hosts both types, it will send the required one to your users.